Avoiding the rebalancing bottleneck for dynamic r-divisions

Develop a method for maintaining an r-division with few holes under O(√r) planarity-preserving edge insertions that avoids the O(r) rebalancing bottleneck, with the goal of improving dynamic planar reachability update bounds.

Background

Several reachability-emulator constructions require the terminals of each piece to lie on O(1) faces. Maintaining this topological condition under boundary-set extensions currently incurs an O(r) rebalancing cost. The paper identifies a more efficient dynamic maintenance procedure for r-divisions with few holes as a concrete unresolved issue that could substantially improve the online algorithm.

References

However, it is currently not clear how to avoid the $(r)$ rebalancing bottleneck when maintaining such $r$-divisions under boundary set extensions.

— Breaking Two Barriers for Dynamic Planar Reachability  (2610.13473 - Karczmarz et al., 9 Oct 2026) in Section 7, Conclusion, paragraph “Online reachability”